Generating a Pseudo-Vandermonde Matrix of the Hermite Polynomial in Python

π‘ Problem Formulation: A Vandermonde matrix is a square matrix with the terms of a geometric progression in each row. When dealing with Hermite polynomials, a pseudo-Vandermonde matrix incorporates the values derived from these polynomials. This article will demonstrate how to generate such a matrix in Python, starting with a sequence of points (e.g., x … Read more

How to Generate a Pseudo Vandermonde Matrix for Hermite Polynomials in Python

π‘ Problem Formulation: A pseudo Vandermonde matrix for Hermite polynomials is computed from a set of points and involves evaluating Hermite polynomials at those points to create the matrix. The input is a float array of point coordinates, e.g., [1.0, 3.5, 5.2]. The desired output is a matrix where each column is the evaluation of … Read more

5 Best Ways to Remove Small Trailing Coefficients from Laguerre Polynomial in Python

π‘ Problem Formulation: When working with Laguerre polynomials in Python, it’s common to encounter situations where small trailing coefficients may be negligible and can be discarded to simplify the polynomial. This article demonstrates multiple techniques to remove such coefficients effectively. Given an input Laguerre polynomial, for example, L(x) = 0.1x^3 + 2.5x^2 + 0.0001x + … Read more

5 Best Ways to Convert a Laguerre Series to a Polynomial in Python

π‘ Problem Formulation: When working with orthogonal polynomials in numerical methods or spectral analysis, one may encounter Laguerre series. These series represent functions as linear combinations of Laguerre polynomials. The task is to convert such a series into a standard polynomial form. For instance, given a Laguerre series defined by coefficients [2, 1, 3], the … Read more

5 Best Ways to Convert a Polynomial to a Laguerre Series in Python

π‘ Problem Formulation: Converting polynomial functions to Laguerre series is a procedure often required in numerical analysis and computational physics. The challenge lies in expressing a given polynomial as a series of Laguerre polynomials. For instance, we want to convert the polynomial p(x) = 3x^2 + 2x + 1 into a series of the form … Read more

5 Best Ways to Add One Legendre Series to Another in Python

π‘ Problem Formulation: In applied mathematics and computational physics, operations on Legendre polynomial series are common. Suppose you have two such series represented in Python and you want to add them together. If series1 has coefficients [1, 3, 5] and series2 has coefficients [2, 4, 6], their addition should yield a new series with coefficients … Read more

5 Best Ways to Subtract One Legendre Series from Another in Python

π‘ Problem Formulation: In computational mathematics, Legendre polynomials are utilized for approximating functions. When handling two Legendre series representing two functions, we might need to compute the difference between them. Suppose we have two series A(x) and B(x); our aim is to find a new series C(x) which represents A(x) – B(x). This article outlines … Read more

5 Best Ways to Multiply a Legendre Series by an Independent Variable in Python

π‘ Problem Formulation: When working with orthogonal polynomials in numerical computations, such as the Legendre polynomials, one often needs to perform various operations on them. A common task is to multiply a Legendre series by an independent variable x, typically for integration, differentiation, or solving differential equations in physical problems. Given a Legendre series a_n, … Read more

5 Best Ways to Multiply One Legendre Series to Another in Python

π‘ Problem Formulation: The task is to find effective methods for multiplying two Legendre series in Python. Legendre series, composed of coefficients corresponding to Legendre polynomials, are used in various numerical computations and approximations. Given two Legendre series, e.g., [a0, a1, a2] and [b0, b1, b2], the goal is to calculate the product series, which … Read more

5 Best Ways to Divide One Legendre Series by Another in Python

π‘ Problem Formulation: When working with Legendre polynomials in numerical and computational applications, it is sometimes necessary to divide one Legendre series by another. This operation can be challenging due to the nature of these polynomials. For instance, if we have two Legendre series represented by P3(x) and P4(x), our goal is to find a … Read more

Generating Pseudo-Vandermonde Matrices with Laguerre Polynomials and XYZ Sample Points in Python

π‘ Problem Formulation: Pseudo-Vandermonde matrices are a generalization of classical Vandermonde matrices, which play a significant role in interpolation problems, least squares fitting, and numeric analysis. Specifically, this article tackles the generation of such matrices where columns correspond to Laguerre polynomials evaluated at given x, y, z sample points. The desired output is a matrix … Read more

Generating a Pseudo Vandermonde Matrix for Laguerre Polynomials in Python

π‘ Problem Formulation: Given an array of floating point coordinates (x, y, z), the objective is to generate a pseudo Vandermonde matrix using the Laguerre polynomials. The Vandermonde matrix is a matrix with the terms of a geometric progression in each row, which, in our case, links to the evaluation of the Laguerre polynomials at … Read more